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arXiv · 1312.6623

On special values of standard L-functions of Siegel cusp eigenforms of genus 3

Anh Tuan Do·Kirill Vankov
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Abstract

We explicitly compute the special values of the standard $L$-function $L(s, F_{12}, \mathrm{St})$ at the critical points $s\in\{-8, -6, -4, -2, 0, 1, 3, 5, 7, 9\}$, where $F_{12}$ is the unique (up to a scalar) Siegel cusp form of degree $3$ and weight $12$, which was constructed by Miyawaki. These values are proportional to the product of the Petersson norms of symmetric square of Ramanujan's $Δ$ and the cusp form of weight $20$ for ${\rm SL}_2(\mathbb{Z})$ by a rational number and some power of $π$. We use the Rankin-Selberg method and apply the Holomorphic projection to compute these values. To our knowledge this is the first example of a standard $L$-function of Siegel cusp form of degree $3$, when the special values can be computed explicitly.

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Anh Tuan Do, Kirill Vankov. 2013-12-23. On special values of standard L-functions of Siegel cusp eigenforms of genus 3. https://arxiv.org/abs/1312.6623

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